Method, system and computer program product for analyzing maintenance operations and assessing the readiness of repairable systems

ABSTRACT

An automated method, system and computer program product for assessing the readiness of a plurality of repairable systems, such as a fleet of aircraft, are provided. In addition to identifying the repairable systems that will be operational, the relative state of readiness of the repairable systems is determined such that the repairable systems that are most likely to successfully complete the designated task can be selected. Additionally, an automated method of analyzing the maintenance operations performed upon a plurality of repairable systems, such as a fleet of aircraft, is provided. In this regard, the relative states of readiness of the repairable systems are determined and maintenance resources are allocated based upon the respective measures of the relative states of readiness of the repairable systems. As such, maintenance operations scheduled for the aircraft that will have the greatest state of readiness upon completion of the maintenance operations can be prioritized.

FIELD OF THE INVENTION

[0001] The present invention relates generally to methods, systems andcomputer program products for analyzing maintenance operations andassessing the readiness of a plurality of repairable systems, such as afleet of aircraft.

BACKGROUND OF THE INVENTION

[0002] Many electromechanical systems must periodically undergomaintenance. For example, an electromechanical system may fail duringoperation so that maintenance is required in order to repair theelectromechanical system and to return the electromechanical system tooperation. Alternatively, an electromechanical system may periodicallyundergo scheduled maintenance in order to reduce the likelihood that theelectromechanical system will fail unexpectedly during operation. Forexample, aircraft generally include a large number of subsystems thatmust be maintained, preferably in accordance with a predefinedmaintenance schedule.

[0003] In some situations, a plurality of electromechanical systems ofthe same type are available for performing a particular function. Forexample, a mission may require that a plurality of aircraft be deployedon a particular day. In order to provide proper resource allocation,such as proper allocation of the aircraft, the availability of theelectromechanical systems must be determined. For example, theavailability of the aircraft on the date of the mission must bedetermined in order to properly identify the aircraft to fly themission. For those electromechanical systems currently undergoingmaintenance or scheduled to undergo maintenance prior to the date ofdeployment, it must therefore be determined if sufficient maintenanceoperations will have been performed such that the electromechanicalsystem will be available.

[0004] In evaluating the status of electromechanical systems followingcompletion of the maintenance operations, the electromechanical systemsare generally considered to be either in the same state as immediatelyprior to the commencement of the maintenance operations, i.e., asbad-as-old, or in a like-new condition, i.e., as good-as-new. Theseassumptions are generally somewhat incorrect, however, since the statusof an electromechanical system following completion of the maintenanceoperations is generally somewhere between as bad-as-old and asgood-as-new. For example, in instances in which the maintenanceoperations repair an electromechanical system that failed duringoperation, the electromechanical system is generally in better shapethan immediately prior to the failure and, as such, will likely operatewithout failure for a longer period of time. However, even after thecompletion of the maintenance operations, the electromechanical systemwill likely not be in like-new condition and will generally be expectedto fail in a somewhat shorter period of time than a newelectromechanical system.

[0005] In order to more accurately determine the status of anelectromechanical system following the completion of maintenanceoperations, a technique that applies a modulated power law process wasdescribed by Scott E. Black, et al., “Statistical Inference for aModulated Power Law Process,” Journal of Quality Technology, Vol. 28,No. 1, pp. 81-90 (January 1996). In this regard, the Black articledescribes a technique in which the relative status of anelectromechanical system following the completion of maintenanceoperations could be determined across a continuum extending from asgood-as-new to as bad-as-old. The contents of the Black article areincorporated by reference herein.

[0006] Unfortunately, conventional resource allocation techniques havenot taken into account the relative status or operability of anelectromechanical system, but have instead, merely focused upon theidentification of those electromechanical systems that will be operableon the date of deployment. With respect to mission requests thatidentify a number of aircraft required to fly a particular mission on apredetermined date, aircraft are selected for the mission from amongthose that will be operational on the predetermined date of the missionwithout any consideration as to the relative status or degree ofreadiness of the aircraft. In determining the availability of theaircraft, a minimum equipment list is typically utilized to identify anumber of subsystems that must be functioning in order for the aircraftto be cleared to fly. As such, aircraft that have the minimum equipmentidentified by the minimum equipment list would be identified as acandidate for the mission without any indication as to the relativedegree of readiness of the aircraft and its respective subsystems.

[0007] Since mission commanders not only wish to begin a mission, but tocomplete the mission as successfully as possible, it would be useful tonot only identify the aircraft that are operational and available on thepredetermined date of the mission, but also to provide some indicationas to the relative degrees of readiness of the aircraft. As such, themission commander could select those aircraft that have the greatestlikelihood of successfully completing the mission without failure of oneor more aircraft subsystems. To date, however, mission commanders arenot provided with information relating to the relative degrees ofreadiness of the aircraft.

SUMMARY OF THE INVENTION

[0008] An automated method, system and computer program product forassessing the readiness of a plurality of repairable systems, such as afleet of aircraft, are therefore provided. In addition to identifyingthe repairable systems that will be operational, the method, system andcomputer program product of this aspect of the present invention alsodetermine the relative state of readiness of the plurality of repairablesystems such that the repairable systems that are most likely tosuccessfully complete the designated task can be selected. Additionally,an automated method of analyzing the maintenance operations performedupon a plurality of repairable systems, such as a fleet of aircraft, isprovided. According to this aspect of the present invention, therelative states of readiness of the plurality of repairable systems aredetermined and maintenance resources are allocated based upon therespective measures of the relative states of readiness of therepairable systems. As such, maintenance operations scheduled for theaircraft that will have the greatest state of readiness upon completionof the maintenance operations can be prioritized.

[0009] According to one aspect of the present invention, an automatedmethod, system and computer program product for assessing the readinessof a plurality of repairable systems are provided. According to thisaspect, at least one system allocation request is received. The systemallocation request typically includes a date and the number of systemsto be allocated to the task. The relative states of readiness of theplurality of repairable systems are then automatically determined. Inthis regard, the relative states of readiness are determined byanalyzing maintenance information associated with the repairable systemsto determine the repairable systems that will be operational on the dateof the requested system allocation. In addition, the relative states ofreadiness are determined by ascertaining respective measures of therelative states of readiness of the repairable systems that will beoperational on the date of the requested system allocation based uponrespective probabilities of failure of the repairable systems followingcompletion of the maintenance operations. In this regard, adetermination of the respective measures of the relative states ofreadiness of the repairable systems on the date of the requested systemallocation may be based upon an intensity function appropriate for thetype of process, such as a modulated power law process, a gamma renewalprocess, a homogenous Poisson process or a power law process, thatdescribes the probability of failure of the repairable systems.

[0010] Based upon the relative states of readiness of the repairablesystems, the systems that will be operational on the date of therequested system allocation will be identified. In addition, therespective measures of the relative states of readiness of therepairable systems identified to be operational on the day the requestedsystem allocation will be provided. As such, in addition to merelyidentifying systems that will be operational on the date of therequested system allocation, the systems having the greatest state ofreadiness on the date of the requested system allocation can beidentified. As such, the systems that are most capable of successfullycompleting the task can be selected to perform the task, therebymaximizing the likelihood of successful completion of the task.According to this aspect of the present invention, a modification of thesystem allocation request may also be proposed in order to increase therelative states of readiness of the systems identified to be operationalon the date of the modified system allocation request in comparison tothe relative states of readiness of the systems identified to beoperational on the date of the original system allocation request.

[0011] According to one advantageous embodiment, an automated method,system and computer program product are provided for assessing thereadiness of a fleet of aircraft, each of which is comprised of aplurality of repairable subsystems. According to this embodiment, atleast one mission request is received that includes a date and a numberof aircraft. Relative states of readiness of a plurality of aircraft arethen determined by automatically analyzing maintenance informationassociated with the plurality of aircraft. Aircraft that will beoperational on the date of the requested mission are then identified andrespective measures of the relative states of readiness of theseaircraft are then provided. In this regard, the aircraft that have thegreatest likelihood of completing the requested mission may beidentified and modifications of the mission request may be proposed inorder to increase the relative states of readiness of the aircraftoperational on the date of the modified mission in comparison to therelative states of readiness of the aircraft operational on the date ofthe requested mission. In order to determine the relative states ofreadiness, the respective probabilities of failure of the aircraftfollowing completion of the maintenance operations may be considered.More particularly, the respective measures of the relative states ofreadiness of the aircraft may be based upon an intensity functionappropriate for the type of process that describes the probability offailure of the aircraft.

[0012] According to another aspect of the present invention, anautomated method of analyzing maintenance operations performed upon aplurality of repairable systems is provided. According to this aspect,maintenance information associated with the plurality of repairablesystems is initially analyzed to determine the relative states ofreadiness of a plurality of repairable systems. Respective measures ofthe relative states of readiness of the repairable systems are thendetermined based upon the respective probabilities of failure of therepairable systems following completion of the maintenance operations.Maintenance resources are then allocated based upon the respectivemeasures of the relative states of readiness of the plurality ofrepairable systems. In this regard, maintenance operations scheduled forthe repairable systems that will have the greatest states of readinessupon completion of the maintenance operations may be prioritized. Asbefore, the determination of the relative states of readiness may bebased upon respective probabilities of failure of the repairable systemsfollowing completion of the maintenance operations. In this regard, therespective measures of the relative states of readiness of therepairable systems may be based upon an intensity function appropriatefor the type of process that describes the probability of failure of therepairable system.

[0013] In one embodiment, the automated method of analyzing maintenanceoperations analyzes maintenance operations performed upon the fleet ofaircraft. In this regard, the maintenance information associated withthe plurality of aircraft is initially analyzed to determine therelative states of readiness of the aircraft upon completion of themaintenance operations scheduled for the aircraft. Respective measuresof the relative states of readiness of the aircraft upon completion ofthe maintenance operations scheduled for the plurality of aircraft arethen provided and maintenance resources are allocated based upon therespective measures of the relative states of readiness of the aircraft.As before, maintenance resources may be allocated by prioritizing themaintenance operations scheduled for the aircraft that will have thegreatest state of readiness upon completion of the maintenanceoperations.

[0014] By not only identifying the repairable systems that will beoperational on the date of deployment, but providing respective measuresof the relative states of readiness of the repairable systems, therepairable systems that will have the greatest likelihood ofsuccessfully completing the task can be selected, thereby maximizing theprobability that the task will be successfully completed. For example,the aircraft that have the greatest likelihood of successfullycompleting a mission can be selected in an educated manner based uponthe relative states of readiness of the aircraft which, in turn, isbased upon the probability of failure or, conversely, success of theaircraft. Additionally, allocation of maintenance resources may beimproved according to another aspect of the present invention byanalyzing the relative states of readiness of the plurality ofrepairable systems, such as a plurality of aircraft, upon the completionof the maintenance operations and then scheduling the maintenanceresources in such a way that the repairable systems that have thegreatest states of readiness are repaired initially.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Having thus described the invention in general terms, referencewill now be made to the accompanying drawings, which are not necessarilydrawn to scale, and wherein:

[0016]FIG. 1 is a block diagram of a system for assessing the readinessof a fleet of aircraft and for analyzing maintenance operationsperformed upon a fleet of aircraft according to an embodiment of thepresent invention;

[0017]FIG. 2 is a flow chart illustrating the operations performed toassess the readiness of a plurality of repairable systems, such as aplurality of aircraft, according to one embodiment of the presentinvention; and

[0018]FIG. 3 is a flow chart illustrating the operations performed toautomatically analyze the maintenance operations performed upon aplurality of repairable systems, such as a plurality of aircraft,according to one embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0019] The present invention now will be described more fullyhereinafter with reference to the accompanying drawings, in whichpreferred embodiments of the invention are shown. This invention may,however, be embodied in many different forms and should not be construedas limited to the embodiments set forth herein; rather, theseembodiments are provided so that this disclosure will be thorough andcomplete, and will fully convey the scope of the invention to thoseskilled in the art. Like numbers refer to like elements throughout.

[0020] Referring now to FIG. 1, a block diagram depicting theinteraction of a method and system 10 according to one embodiment of thepresent invention with a conventional maintenance management system 12is depicted. While the system and method of this embodiment of thepresent invention are illustrated and will be described in conjunctionwith the analysis of the maintenance operations performed upon a fleetof aircraft, the system and method can be utilized in conjunction with awide variety of other repairable systems, typically electromechanicalsystems.

[0021] As shown in FIG. 1 and in block 40 of FIG. 2, the method andsystem 10 of the present invention receive a system allocation request,such as a mission request. A mission request may include a variety ofparameters depending upon the mission but typically includes the numberof aircraft required for the mission, a projected date of the mission,an indication as to whether the mission may be postponed, the relativeurgency or ranking of the mission and the like. Moreover, the missionrequest may define any particular characteristics, such as the minimumequipment, that will be required of the aircraft selected for themission. Although not necessary for the practice of the presentinvention, the mission request(s) may be compiled in a queue 14 andprovided in various fashions, such as in a chronological fashion orbased upon the relative urgency or ranking of the missions.

[0022] Based upon the mission request, the method and system 10 of thepresent invention will analyze the current status of the fleet ofaircraft and provide a response indicating the aircraft that will beavailable to fly the mission as well as an indication of the relativestates of readiness of the aircraft as shown generally by FIG. 2. Inorder to provide this information, the system and method of the presentinvention interfaces with a conventional maintenance management system12. The maintenance management system typically provides dynamicresource management by identifying candidate maintenance operations thatshould be performed upon an aircraft. In this regard, the maintenancemanagement system may determine candidate maintenance actions based uponpredefined maintenance requirements and procedures as well asinformation relating to the aircraft, such as the age, the flight timeand prior maintenance activities performed upon the aircraft. As such,the maintenance management system can identify aircraft that shouldundergo scheduled maintenance of one or more subsystems. As indicated byFIG. 2, the maintenance management system may include or communicatewith a database 18 that includes a wide variety of data relating to theaircraft, such as the aircraft configuration by tail number, and themaintenance history of the aircraft as well as the available andscheduled maintenance resources. In addition, the maintenance managementsystem may receive input from maintenance personnel who arrange forotherwise unscheduled maintenance operations, such as to repairparticular subsystems that have exhibited a fault. The maintenancepersonnel can provide this information in various manners, such as via amaintenance terminal 16 as illustrated in FIG. 1. For example, themaintenance management system and associated maintenance terminal may bethe maintenance systems provided by Aero Info, Inc. of Sonora, Calif.

[0023] Based upon the candidate maintenance operations, the maintenancemanagement system 12 will schedule the aircraft for maintenance basedupon the resources available and prior commitments for those resources.As such, the maintenance management system serves a scheduling function.In addition, the maintenance management system will monitor maintenanceactivity and update the maintenance records associated with the aircraftas various maintenance operations are performed and completed. Basedupon the data maintained by the maintenance management system, themethod and system 10 of the present invention can therefore determinethe aircraft that will be operational on the projected date of amission. See blocks 42 and 44. In this regard, the aircraft that will beavailable for a mission generally includes aircraft that are notcurrently scheduled to undergo maintenance as well as those aircraftundergoing maintenance that will be completed by the projected date ofthe mission.

[0024] In addition to identifying the aircraft that will be operationalon the projected date of the mission, the method and system 10 of thepresent invention also provide an indication or measure of the relativestates of readiness of the aircraft identified to be capable ofperforming the requested mission. See block 46. In this regard, thesystem and method of the present invention are designed to determine therelative states of readiness based upon the respective probabilities offailure or, conversely, success of the aircraft upon completion of themaintenance operations. In comparison to conventional techniques thatconsider a repaired system as being as good-as-new or as bad-as-old, thesystem and method of the present invention determines the relativestates of readiness across the continuum extending from as good-as-newto as bad-as-old. As such, the system and method of the presentinvention provide a more accurate indication of the relative states ofreadiness of the aircraft such that an aircraft can be assigned to themission in a more educated fashion with a greater degree of confidencethat the aircraft will be able to successfully complete the missionwithout failure of any of the subsystems.

[0025] The method and system 10 of the present invention can beconfigured to utilize various probabilistic techniques to providerespective measures of the relative states of readiness of the aircraft,or other repairable systems. In one advantageous embodiment, the methodand system determine the type of process that describes the probabilityof failure of the aircraft and, based upon the type of process, providethe measures of the relative states of readiness of the aircraft. Forexample, measures of the relative states of readiness of the aircraftare typically based upon the intensity function associated with the typeof process that describes the probability of failure of the aircraft.However, other measures, such as parameters associated with the type ofprocess that describes the probability of failure of the aircraft, maybe utilized as described below.

[0026] Initially, the method and system 10 analyzes the relative statesof readiness of the aircraft based upon a modulated power law process,although the method and system may subsequently determine that theprocess that describes the probability of failure of an aircraft isactually a more specific form of the modulated power law process, suchas a gamma renewal process, a homogenous Poisson process or a power lawprocess, thereby permitting the measures of the relative states ofreadiness of the aircraft to be refined. As known to those skilled inthe art, and as described by the Black article, the modulated power lawprocess is a three-parameter stochastic point process model that can beused to describe the failure times of a repairable system. As also knownto those skilled in the art, the three-parameters are kappa, theta andbeta. As described hereinafter, maximum likelihood estimates areobtained for each of the three model parameters. Thereafter, confidenceintervals and hypothesis tests are performed for the parameters in orderto provide a degree of reliability for the estimated parameters.

[0027] In order to obtain the maximum likelihood estimates for thethree-parameters, a joint probability density function is initiallydefined for the first n failure times t₁>t₂> . . . >t_(n) of an aircraftin response to shocks or events, such as flights, as follows:$\begin{matrix}{{f\left( {t_{1},t_{2},\ldots \quad,t_{n}} \right)} = {\left\{ {\prod\limits_{i - 1}^{n}\quad {{u\left( t_{i} \right)}\left\lbrack {{U\left( t_{i} \right)} - {U\left( t_{i - 1} \right)}} \right\rbrack}^{k - 1}} \right\} \times \frac{\exp \left\lbrack {- {U\left( t_{i} \right)}} \right\rbrack}{\left\lbrack {\Gamma (k)} \right\rbrack^{n}}}} & (1)\end{matrix}$

[0028] wherein u(t) is the intensity function of a nonhomogenous Poissonprocess that is defined as follows: $\begin{matrix}{{{u(t)} = {\frac{\beta}{\theta}\left( \frac{t}{\theta} \right)^{\beta - 1}}},{t > 0}} & (2)\end{matrix}$

[0029] and wherein U(t) is the expected number of shocks before time tand is defined as

U(t)=∫₀ ^(t) u(x)dx.  (3)

[0030] and further wherein according to a nonhomogenous Poisson processa failure occurs not at every shock but at every κ^(th) shock with κbeing a positive value. For example, if kappa=4, then every fourth shockwould cause a failure.

[0031] Kappa is a measure of the improvement effected by a repair suchthat for larger values of kappa, the improvement provided by the repairwill be more significant. Conversely, beta is a measure of theimprovement or deterioration of a system over the course of the life ofthe system. In addition, theta is a scaling parameter that is typicallyutilized to change units, such as from a week to a month.

[0032] For the nonhomogenous Poisson processes having the power lawprocess intensity set forth by equation (2), the log likelihood functionof the joint probability density function of equation (1) can bepresented as follows: $\begin{matrix}{l\left( {\theta,\beta,{\kappa {{t_{1},t_{2},\ldots \quad,{t_{n} = {{\ln \quad {L\left( {\theta,\beta,\kappa} \right)}} = {{- \left( \frac{t_{n}}{\theta} \right)^{\beta}} + {n\quad \ln \quad \beta} - {n\quad \ln \quad {\Gamma (\kappa)}} - {n\quad {\beta\kappa}\quad \ln \quad \theta} + {\left( {\beta - 1} \right){\sum\limits_{i = 1}^{n}\quad {\ln \quad t_{1}}}} + {\left( {\kappa - 1} \right){\sum\limits_{i = 1}^{n}\quad {\ln \left( {t_{i}^{\beta} - t_{i - 1}^{\beta}} \right)}}}}}}}}}} \right.} & (4)\end{matrix}$

[0033] In order to determine the maximum likelihood estimates of thethree parameters, beta, theta and kappa, the partial derivatives of thefunction l set forth by equation (4) are taken and set equal to zero asfollows: $\begin{matrix}{\frac{\partial l}{\partial\theta} = {0 = {{\frac{\beta}{\theta}\left( \frac{t_{n}}{\theta} \right)^{\beta}} - \frac{n\quad {\beta\kappa}}{\theta}}}} & (5) \\{\frac{\partial l}{\partial\beta} = {0 = {{{- \left( \frac{t_{n}}{\theta} \right)^{\beta}}\ln \frac{t_{n}}{\theta}} + \frac{n}{\beta} - {n\quad {\kappa ln\theta}} + {\sum\limits_{i = 1}^{n}\quad {\ln \quad t}} + {\left( {\kappa - 1} \right){\sum\limits_{i = 1}^{n}\quad \frac{{t_{i}^{\beta}\ln \quad t_{i}} - {t_{i - 1}^{\beta}\ln \quad t_{i - 1}}}{t_{i}^{b} - t_{i}^{\beta} - 1}}}}}} & (6) \\{\frac{\partial l}{\partial\kappa} = {0 = {{n\quad {\psi (\kappa)}} - {n\quad {\beta ln\theta}} + {\sum\limits_{i = 1}^{n}\quad {{\ln \left( {t_{i}^{\beta} - t_{i - 1}^{\varphi}} \right)}.}}}}} & (7)\end{matrix}$

[0034] By solving for beta, theta and kappa, the maximum likelihoodestimate for each parameter is obtained. Various methods can be utilizedto solve for these parameters including the Newton-Raphson method andthe Nelder-Mead simplex algorithm. In one advantageous embodiment,however, a combination of the Nelder-Mead simplex algorithm and theNewton-Raphson method provides an efficient and reliable algorithm fordetermining the maximum likelihood estimates of the three parameters. Inthis regard, several iterations of the Nelder-Mead algorithm areperformed, such as about 50 iterations, with the last point of thisiterative scheme being utilized as the initial point of theNewton-Raphson method. Other details regarding this solution techniqueare provided by the Black article.

[0035] Based upon the maximum likelihood estimates for the threeparameters, the relative states of readiness of the aircraft can beapproximated. In this regard, kappa is a measure of the improvementeffected by a repair, while beta is a measure of the improvement ordeterioration of a system over the course of its life. Since the maximumlikelihood estimates of the three parameters are estimates, the methodand system 10 of the present invention may also determine confidenceintervals and/or hypothesis tests for the parameters in order to developgreater confidence that the maximum likelihood estimates are accurate.The confidence intervals are typically derived using asymptoticapproaches. The asymptotic distribution of the estimate for each of thethree parameters has a multivariate normal distribution with a mean μdefined as μ=[θ,β,κ,] and a variance defined by a covariance matrix[J(θ,β,κ)]⁻¹. In this regard, the J matrix is comprised of secondpartial derivatives of the log likelihood function set forth in equation(4) and is represented as follows: $\begin{matrix}{{J\left( {\theta,\beta,\kappa} \right)} = \begin{bmatrix}\frac{\partial^{2}l}{\partial\theta^{2}} & \frac{\partial^{2}l}{{\partial\theta}{\partial\beta}} & \frac{\partial^{2}l}{{\partial\theta}{\partial\kappa}} \\\frac{\partial^{2}l}{{\partial\theta}{\partial\beta}} & \frac{\partial^{2}l}{\partial\beta^{2}} & \frac{\partial^{2}l}{{\partial\beta}{\partial\kappa}} \\\frac{\partial^{2}l}{{\partial\theta}{\partial\kappa}} & \frac{\partial^{2}l}{{\partial\beta}{\partial\kappa}} & \frac{\partial^{2}l}{\partial\kappa^{2}}\end{bmatrix}} & (8)\end{matrix}$

[0036] Based upon the covariance matrix, the confidence intervals foreach of the parameters are defined as follows:

{circumflex over (θ)}±z_(α/2){square root}{square root over ((1,1) entryin [J)}({circumflex over (θ)},{circumflex over (β)},{circumflex over(κ)})]⁻¹  (9)

{circumflex over (β)}±z_(α/2){square root}{square root over ((2,2) entryin [J)}({circumflex over (θ)},{circumflex over (β)},{circumflex over(κ)})]⁻¹  (10)

{circumflex over (κ)}±z_(α/2){square root}{square root over ((3,3) entryin [J)}({circumflex over (θ)},{circumflex over (β)},{circumflex over(κ)})]⁻¹  (11)

[0037] whenever {circumflex over (θ)}, {circumflex over (β)} and{circumflex over (κ)} are the maximum likelihood estimates of theta,beta and kappa, respectively. As known to those skilled in the art, zindicates that the parameters have a standard normal distribution andcan be obtained from a look-up table or the like. See, for example, LeeJ. Bain and Max Engelhardt, Introduction to Probability and MathematicalStatistics, P. W. S. Kent (1987). In addition, alpha is based upon theconfidence interval. For example, in instances in which the confidenceintervals are the 95% confidence intervals such that there is a 95%likelihood that the parameter will fall within the range defined by theinterval, alpha will equal 0.05. Likewise, a 90% confidence intervalwould have alpha equal to 0.10 and a 99% confidence interval would havealpha equal to 0.01.

[0038] Based upon the confidence intervals, the degree of reliability ofthe maximum likelihood estimates for the parameters can be determinedwith parameters having larger confidence intervals somewhat less certainthan parameters having smaller confidence intervals. In order to providea further indication of the reliability of the maximum likelihoodestimates, the method and system 10 of the present invention may alsotest several hypotheses. In this regard, there are several special andmore specific cases of the modulated power law process, such as ininstances in which kappa equals 1 in which the modulated power lawprocess reduces to the power law process, instances in which beta equals1 in which the modulated power law process reduces to the gamma renewalprocess and instances in which both beta and kappa equal 1 in which themodulated power law process reduces to the homogeneous Poisson process.In order to determine the proper process for modeling the probability offailure of the aircraft or other repairable system, the likelihood ratiotest statistic is determined for each instance as follows:$\begin{matrix}{\lambda = \frac{\max\limits_{{({\theta,\beta,\kappa})} \in H_{0}}{l\left( {\theta,\beta,\kappa} \right)}}{\max\limits_{({\theta,\beta,\kappa})}{l\left( {\theta,\beta,\kappa} \right)}}} & (12)\end{matrix}$

[0039] wherein (θ,β,κ)εH₀ represents those values of (θ,β,κ) thatsatisfy the null hypothesis, i.e., κ=1, β=1 or κ=β=1. For all threehypothesis, the denominator is the maximum of the log likelihoodfunction with no restrictions on the parameters. In other words, thedenominator is the log likelihood function evaluated at the maximumlikelihood estimates {circumflex over (θ)}, {circumflex over (β)},{circumflex over (κ)}, as set forth below: $\begin{matrix}{{\max\limits_{({\theta,\beta,\kappa})}{l\left( {\theta,\beta,\kappa} \right)}} = {l\left( {\hat{\theta},\hat{\beta},\hat{\kappa}} \right)}} & (13)\end{matrix}$

[0040] As will be apparent, the numerator will depend on whichhypothesis is being tested. In order to test the likelihood of κ=1 inwhich the modulated power law process reduces to the power law processversus the likelihood of κ≠1, the numerator is the maximum of the loglikelihood function under the restriction that κ=1. But if κ=1 then thelog likelihood function is just the log likelihood function for thepower law process, which achieves its maximum when: $\begin{matrix}{{\hat{\beta}}_{1} = {\frac{n}{\sum\limits_{i = 1}^{n - 1}\quad {\ln \left( {t_{n}/t_{i}} \right)}}\quad {and}}} & (14) \\{{\hat{\theta}}_{1} = {\frac{t_{n}}{n^{1/{\hat{\beta}}_{1}}}.}} & (15)\end{matrix}$

[0041] The numerator is thus $\begin{matrix}{{\max\limits_{{({\theta,\beta,\kappa})} \in H_{0}}{l\left( {\theta,\beta,\kappa} \right)}} = {{l\left( {{\hat{\theta}}_{1},{\hat{\beta}}_{1},1} \right)}.}} & (16)\end{matrix}$

[0042] Thus the likelihood ratio test statistic for testing thehypothesis in which κ=1 is: $\begin{matrix}{\lambda_{1} = \frac{l\left( {{\hat{\theta}}_{1},{\hat{\beta}}_{1},1} \right)}{l\left( {\hat{\theta},\hat{\beta},\hat{\kappa}} \right)}} & (17)\end{matrix}$

[0043] Typically, the null hypothesis that κ=1 is rejected if−2lnλ₁>χ_(1-a) ²(1) wherein χ_(1-a) ² is a chi-square distribution asknown to those skilled in the art and the (1) represents the number ofrestrictions on the parameters. However, it should be understood thatthe method and system may utilize other tests or thresholds fordetermining if a null hypothesis is to be accepted or rejected.

[0044] In order to test the hypothesis in which β is equal to one andthe modulated power law process reduces to a gamma renewal process inwhich the system does not experience deterioration throughout itslifetime, although it may experience degradation during the timesbetween failures, i.e. a repaired unit is in exactly the same conditionas a new unit, the denominator in the likelihood ratio test statistic isthe log likelihood function evaluated at the maximum likelihoodestimates, as shown in equation (13). The numerator in the likelihoodratio statistic is equal to: $\begin{matrix}{{\max\limits_{{({\theta,\beta,\kappa})} \in H_{0}}{l\left( {\theta,\beta,\kappa} \right)}} = {l\left( {{\hat{\theta}}_{2},{\hat{\beta}}_{21},{\hat{\kappa}}_{2}} \right)}} & (18)\end{matrix}$

[0045] wherein {circumflex over (θ)}₂ and {circumflex over (κ)}₂ are theestimates of θ and κ under the assumption that β=1. Therefore, theparameters θ and κ of a gamma renewal process must be estimated. Themaximum likelihood estimates of θ and κ do not have closed formexpressions and must be approximated by a numerical procedure. In thisregard, let X₁=T₁, X₂=T₂−T₁, . . . , X_(n)=T_(n)−T_(n-1) denote thetimes between failures. Then X₁, X₂, . . . , X_(n) form a random sampleof size n from the gamma distribution with parameters θ and κ.Differentiating the likelihood function and setting the results equal tozero leads to the following equations:

ln{circumflex over (κ)} ₂−Γ′({circumflex over (κ)}₂)/Γ({circumflex over(κ)}₂)−ln({overscore (x)}/{tilde over (x)})=0  (19)

[0046] and $\begin{matrix}{{\hat{\theta}}_{2} = \frac{\overset{\_}{x}}{{\hat{\kappa}}_{2}}} & (20)\end{matrix}$

[0047] wherein {overscore (x)} is the sample median. An iterative methodsuch as Newton's method is required to solve the first equation for κ₂.Once κ₂ is obtained, the equation (2) gives a closed form expression for{circumflex over (θ)}₂. See the Black article for a further discussionof the iterative method.

[0048] Given {circumflex over (θ)}₂ and {circumflex over (κ)}₂, thesevalues may be substituted into the log likelihood function to obtain:$\begin{matrix}{\lambda_{2} = \frac{l\left( {{\hat{\theta}}_{2},1,{\hat{\kappa}}_{2}} \right)}{l\left( {\hat{\theta},\hat{\beta},\hat{\kappa}} \right)}} & (21)\end{matrix}$

[0049] Again, the null hypothesis that β=1 is generally rejected if−2lnλ₂>χ_(1-α) ² (1). However, it should be understood that the systemand method may employ other criteria in determining the acceptability ofthe null hypothesis, if so desired.

[0050] Finally, the hypothesis in which κ=1, β=1 can be tested in whichthe modulated power law process reduces to the homogeneous Poissonprocess. If this hypothesis is true, the times between failure areindependent and identically distributed exponential random variables, sothe maximum likelihood estimate of θ is: $\begin{matrix}{{\hat{\theta}}_{3} = {\frac{t_{n}}{n}.}} & (22)\end{matrix}$

[0051] The likelihood ratio test statistic is thus $\begin{matrix}{\lambda_{3} = \frac{l\left( {{\hat{\theta}}_{3},1,1} \right)}{l\left( {\hat{\theta},\hat{\beta},\hat{\kappa}} \right)}} & (23)\end{matrix}$

[0052] Once again, the hypothesis is typically rejected if−2lnλ₃>χ_(1-α) ² (2), although other tests or thresholds may be utilizedif desired.

[0053] Based upon the hypothesis testing, the method and system obtainadditional information relating to the maximum likelihood estimates. Forexample, the hypothesis testing may determine that the modulated powerlaw process can be reduced to a power law process, a gamma renewalprocess or a homogeneous Poisson process. In these instances, theanticipated behavior of the repairable system, such as the probabilityof failure of the repairable system, can also be determined with greaterreliability and precision.

[0054] While various measures of the relative states of readiness of anaircraft or other repairable system may be generated, the method andsystem 10 of one embodiment determine the probability of failure of therepairable system based upon an analysis of the complete intensityfunction appropriate for the process that best models the repairablesystem. In instances in which the hypothesis testing does not lead tothe repairable system being modeled as one of the more specificprocesses, such as a power law process, a gamma renewal process or ahomogeneous Poisson process, a measure of the relative state ofreadiness of an aircraft or other repairable system is generated basedupon an analysis of the complete intensity function of a modulated powerlaw process. In this regard, the complete intensity function of amodulated power law process is defined as follows: $\begin{matrix}{u\left( {{t\left. t_{n - 1} \right)} = \left\{ {\int_{t}^{\infty}{\left( \frac{x}{t} \right)^{\beta - 1}\left( \frac{x^{\beta} - t_{n - 1}^{\beta}}{t^{\beta} - t_{n - 1}^{\beta}} \right)^{\kappa - 1} \times \exp \left\{ {{- \left( \frac{x}{\theta} \right)^{\beta}} + \left( \frac{t}{\theta} \right)^{\beta}} \right\} {x}}} \right\}^{- 1}} \right.} & (24)\end{matrix}$

[0055] Since the solution of the foregoing equation defines theprobability of failure of the repairable system, the solution alsodefines the relative state of readiness of the repairable system. Ininstances in which the process that best models the repairable system isa power law process, gamma renewal process or a homogenous Poissonprocess, the relative state of readiness of the repairable system may bedetermined in an analagous manner by solving the complete intensityfunction associated with the respective type of process. In this regard,the complete intensity functions associated with these other morespecific types of processes are well known to those skilled in the art.

[0056] As mentioned above, the method and system 10 may determine ameasure of the relative state of readiness of a repairable system inother manners without departing from the spirit and scope of the presentinvention. For example, in instances in which the values of beta for anumber of repairable systems are substantially equal, the respectivevalues of kappa for the repairable systems may be directly utilized toprovide a measure of the relative state of readiness of a repairablesystems.

[0057] In addition to merely identifying the aircraft that will beoperational on the date of the requested mission such as by determiningthe aircraft for which maintenance operations will be completed by thedate of the upcoming mission, the system and method can thereforeprovide respective measures of the relative states of readiness of theaircraft identified to be operational on the date of the requestedmission. See block 48. As described above, various measures of therelative states of readiness of the aircraft can be provided with anobjective of selecting the aircraft having the greatest likelihood ofcompleting the mission based upon the identification of the aircrafthaving greatest states of readiness in order to maximize the probabilityof success of the mission.

[0058] The system and method of the present invention may also propose amodification of the mission request if the proposed modification wouldincrease the relative states of readiness of the aircraft operational onthe date of the modified mission in comparison to the relative states ofreadiness of the aircraft operational on the date of the requestedmission. See block 50. In this regard, if the mission request indicatesthat the date of the mission may be postponed and/or if the urgency orimportance of the mission is relatively low, the system and method maydetermine if aircraft that are undergoing maintenance and that will notbe ready on the initial date of the requested mission will have agreater state of readiness upon the completion of the maintenanceoperations than the aircraft that will be ready to fly upon the date ofthe original mission. If so, the system and method may propose to delaythe mission such that the maintenance operations for these additionalaircraft may be completed in order for the aircraft that are scheduledfor the mission to greater state of readiness, on average, than if themission were performed as originally scheduled. The mission commandermay then opt to delay the mission in order to improve the overall stateof readiness of the aircraft or may elect to continue with the missionas originally planned. While one example has been provided, it should beunderstood that the system and method can propose a variety of othermodifications to the mission request in order to increase the relativestates of readiness of the aircraft, if so desired.

[0059] As shown in FIG. 3, a method is also provided according toanother aspect of the present invention for analyzing maintenanceoperations performed upon the fleet of aircraft or other repairablesystems. In this regard, the maintenance information associated with theaircraft is analyzed to determine the relative states of readiness ofthe aircraft upon completion of the maintenance operations scheduled forthe aircraft. See blocks 60 and 62 of FIG. 3. As described above,respective measures of the relative states of readiness of the aircraftupon completion of the maintenance operations scheduled for the aircraftare provided. See block 64. As described above, measures of theprobability of failure may be provided for each aircraft based upon, forexample, an analysis of the complete intensity function appropriate forthe type of process that best models the aircraft, in order to providean indication of the relative states of readiness of the aircraft.According to this aspect of the present invention, the system and methodmay allocate the maintenance resources such as maintenance personnel,equipment, floor space and the like based upon the respective measuresof the relative states of readiness of the plurality of aircraft. Seeblock 66. For example, the maintenance resources may be prioritized inorder to complete the maintenance operations ahead of time for theaircraft that will have the greatest states of readiness upon completionof the maintenance operation. As such, the aircraft that will have thegreatest states of readiness can be returned to the fleet for activeduty in the shortest period of time.

[0060] The system 10 of the present invention is typically embodied by aprocessing element 20 and an associated memory device, both of which arecommonly comprised by a computer or the like. As such, the system of thepresent invention generally operates under control of a computer programproduct according to another aspect of the present invention. Thecomputer program product for performing the contingent claim valuationincludes a computer-readable storage medium, such as the nonvolatilestorage medium, and computer-readable program code portions, such as aseries of computer instructions, embodied in the computer-readablestorage medium.

[0061] In this regard, FIGS. 1-3 are block diagrams and flowcharts ofmethods, systems and program products according to the invention. Itwill be understood that each block or step of the flowchart, andcombinations of blocks in the flowchart, can be implemented by computerprogram instructions. These computer program instructions may be loadedonto a computer or other programmable apparatus to produce a machine,such that the instructions which execute on the computer or otherprogrammable apparatus create means for implementing the functionsspecified in the flowchart block(s) or step(s). These computer programinstructions may also be stored in a computer-readable memory that candirect a computer or other programmable apparatus to function in aparticular manner, such that the instructions stored in thecomputer-readable memory produce an article of manufacture includinginstruction means which implement the function specified in theflowchart block(s) or step(s). The computer program instructions mayalso be loaded onto a computer or other programmable apparatus to causea series of operational steps to be performed on the computer or otherprogrammable apparatus to produce a computer implemented process suchthat the instructions which execute on the computer or otherprogrammable apparatus provide steps for implementing the functionsspecified in the flowchart block(s) or step(s).

[0062] Accordingly, blocks or steps of the flowchart supportcombinations of means for performing the specified functions,combinations of steps for performing the specified functions and programinstruction means for performing the specified functions. It will alsobe understood that each block or step of the flowchart, and combinationsof blocks or steps in the flowchart, can be implemented by specialpurpose hardware-based computer systems which perform the specifiedfunctions or steps, or combinations of special purpose hardware andcomputer instructions.

[0063] Therefore, the method and system 10 of the present inventionpermit the relative states of readiness of a fleet of aircraft to bedetermined in an automated fashion such that more educated decisions canbe made during the process of selecting the aircraft to fly a mission soas to maximize the likelihood of the aircraft successfully completingthe mission. Similarly, an automated method of analyzing the maintenanceoperations performed upon a fleet of aircraft is provided such that theallocation of the maintenance resources can be performed in an educatedmanner based upon the respective measures of the relative states ofreadiness of the aircraft, thereby permitting the aircraft having thegreatest states of readiness to be returned to service in the shortestperiod of time. While the system, method, and computer program productof the present invention have been primarily described above in thecontext of maintenance operations performed upon a fleet of aircraft,the method, system and computer program product of the present inventioncan likewise analyze the maintenance operations performed upon any of awide variety of repairable systems in order to optimize resourceallocation and the scheduling of maintenance operations.

[0064] Many modifications and other embodiments of the invention willcome to mind to one skilled in the art to which this invention pertainshaving the benefit of the teachings presented in the foregoingdescriptions and the associated drawings. Therefore, it is to beunderstood that the invention is not to be limited to the specificembodiments disclosed and that modifications and other embodiments areintended to be included within the scope of the appended claims.Although specific terms are employed herein, they are used in a genericand descriptive sense only and not for purposes of limitation.

That which is claimed:
 1. An automated method of assessing readiness ofa fleet of aircraft comprising: receiving at least one mission requestincluding a date and a number of aircraft; automatically determiningrelative states of readiness of a plurality of aircraft of the fleet,wherein determining the relative states of readiness comprisesautomatically analyzing maintenance information associated with theplurality of aircraft to determine the relative states of readiness ofthe plurality of aircraft on the date of the requested mission; andidentifying aircraft that will be capable of performing the requestedmission and providing respective measures of the relative states ofreadiness of the aircraft identified to be capable of performing therequested mission.
 2. A method according to claim 1 wherein identifyingaircraft that will be capable of performing the requested missioncomprises identifying the aircraft having the greatest likelihood ofcompleting the requested mission.
 3. A method according to claim 1further comprising proposing a modification of the mission request inorder to increase the relative states of readiness of the aircraftcapable of performing the modified mission in comparison to the relativestates of readiness of the aircraft capable of performing the requestedmission.
 4. A method according to claim 1 wherein determining therelative states of readiness further comprises determining the relativestates of readiness based upon respective probabilities of failure ofthe aircraft following completion of the maintenance operations.
 5. Amethod according to claim 4 wherein determining the relative states ofreadiness based upon respective probabilities of failure of the aircraftfollowing completion of the maintenance operations comprises determiningthe relative states of readiness based upon an intensity functionappropriate for the type of process that describes the probability offailure of the aircraft.
 6. A computer program product for assessingreadiness of a fleet of aircraft, the computer program productcomprising a computer-readable storage medium having computer-readableprogram code portions stored therein, the computer-readable program codeportions comprising: a first executable portion for receiving at leastone mission request including a date and a number of aircraft; a secondexecutable portion for automatically determining relative states ofreadiness of a plurality of aircraft of the fleet, wherein said secondexecutable portion is also capable of automatically analyzingmaintenance information associated with the plurality of aircraft todetermine the relative states of readiness of the plurality of aircrafton the date of the requested mission; and a third executable portion foridentifying aircraft that will be capable of performing the requestedmission, wherein said third executable portion is also capable ofproviding respective measures of the relative states of readiness of theaircraft identified to be capable of performing the requested mission.7. A computer program product according to claim 6 wherein said thirdexecutable portion is further capable of identifying the aircraft havingthe greatest likelihood of completing the requested mission.
 8. Acomputer program product according to claim 6 further comprising afourth executable portion for proposing a modification of the missionrequest in order to increase the relative states of readiness of theaircraft capable of performing the modified mission in comparison to therelative states of readiness of the aircraft capable of performing therequested mission.
 9. A computer program product according to claim 6wherein said second executable portion is further capable of determiningthe relative states of readiness based upon respective probabilities offailure of the aircraft following completion of the maintenanceoperations.
 10. A computer program product according to claim 9 whereindetermining the relative states of readiness based upon respectiveprobabilities of failure of the aircraft following completion of themaintenance operations comprises determining the relative states ofreadiness based upon an intensity function appropriate for the type ofprocess that describes the probability of failure of the aircraft.
 11. Asystem for automatically assessing readiness of a fleet of aircraftcomprising a processing element capable of receiving at least onemission request including a date and a number of aircraft, saidprocessing element also capable of automatically determining relativestates of readiness of a plurality of aircraft of the fleet based uponan automated analysis of maintenance information associated with theplurality of aircraft to determine the relative states of readiness ofthe plurality of aircraft on the date of the requested mission, andwherein said processing element is further capable of identifyingaircraft that will be capable of performing the requested mission andproviding respective measures of the relative states of readiness of theaircraft identified to be capable of performing the requested mission.12. A system according to claim 11 wherein said processing element isfurther capable of identifying the aircraft having the greatestlikelihood of completing the requested mission.
 13. A system accordingto claim 11 wherein said processing element is further capable ofproposing a modification of the mission request in order to increase therelative states of readiness of the aircraft capable of performing themodified mission in comparison to the relative states of readiness ofthe aircraft capable of performing the requested mission.
 14. A systemaccording to claim 11 wherein said processing element is further capableof determining the relative states of readiness based upon respectiveprobabilities of failure of the aircraft following completion of themaintenance operations.
 15. A system according to claim 14 wherein saidprocessing element is further capable of determining the relative statesof readiness based upon respective probabilities of failure of theaircraft following completion of the maintenance operations bydetermining the relative states of readiness based upon an intensityfunction appropriate for the type of process that describes theprobability of failure of the aircraft.
 16. An automated method ofanalyzing maintenance operations performed upon a fleet of aircraftcomprising: automatically analyzing maintenance information associatedwith the plurality of aircraft to determine relative states of readinessof the plurality of aircraft upon completion of the maintenanceoperations scheduled for the plurality of aircraft; providing respectivemeasures of the relative states of readiness of the plurality ofaircraft upon completion of the maintenance operations scheduled for theplurality of aircraft; and allocating maintenance resources based uponthe respective measures of the relative states of readiness of theplurality of aircraft.
 17. A method according to claim 16 whereinallocating maintenance resources comprises prioritizing the maintenanceoperations scheduled for the aircraft that will have the greatest stateof readiness upon completion of the maintenance operations.
 18. A methodaccording to claim 16 wherein determining the relative states ofreadiness comprises determining the relative states of readiness basedupon respective probabilities of failure of the aircraft followingcompletion of the maintenance operations.
 19. A method according toclaim 18 wherein determining the relative states of readiness based uponrespective probabilities of failure of the aircraft following completionof the maintenance operations comprises determining the relative statesof readiness based upon an intensity function appropriate for the typeof process that describes the probability of failure of the aircraft.20. An automated method of assessing readiness of a plurality ofrepairable systems comprising: receiving at least one system allocationrequest including a date and a number of systems to be allocated; andautomatically determining relative states of readiness of the pluralityof repairable systems, wherein determining the relative states ofreadiness comprises: analyzing maintenance information associated withthe plurality of repairable systems to determine the repairable systemsthat will be operational on the date of the requested system allocation;and determining respective measures of the relative states of readinessof the repairable systems that will be operational on the date of therequested system allocation based upon respective probabilities offailure of the repairable systems following completion of themaintenance operations.
 21. A method according to claim 20 furthercomprising identifying systems that will be operational on the date ofthe requested system allocation.
 22. A method according to claim 21further comprising providing the respective measures of the relativestates of readiness of the repairable identified to be operational onthe date of the requested system allocation.
 23. A method according toclaim 21 wherein identifying systems that will be operational on thedate of the requested system allocation comprises identifying thesystems having the greatest state of readiness on the date of therequested system allocation.
 24. A method according to claim 21 furthercomprising proposing a modification of the system allocation request inorder to increase the relative states of readiness of the systemsidentified to be operational on the date of the modified systemallocation request in comparison to the relative states of readiness ofthe systems identified to be operational on the date of the originalsystem allocation request.
 25. A method according to claim 21 whereindetermining the respective measures of the relative states of readinessof the repairable systems comprises determining respective measures ofthe relative states of readiness of the repairable systems on the dateof the requested system allocation based upon an intensity functionappropriate for the type of process that describes the probability offailure of the repairable systems.
 26. An automated method of analyzingmaintenance operations performed upon a plurality of repairable systemscomprising: analyzing maintenance information associated with theplurality of repairable systems to determine relative states ofreadiness of the plurality of repairable systems; determining respectivemeasures of the relative states of readiness of the repairable systemsbased upon respective probabilities of failure of the repairable systemsfollowing completion of the maintenance operations; and allocatingmaintenance resources based upon the respective measures of the relativestates of readiness of the plurality of repairable systems.
 27. A methodaccording to claim 26 wherein allocating maintenance resources comprisesprioritizing the maintenance operations scheduled for the repairablesystems that will have the greatest state of readiness upon completionof the maintenance operations.
 28. A method according to claim 26wherein determining the relative states of readiness comprisesdetermining the relative states of readiness based upon respectiveprobabilities of failure of the repairable systems following completionof the maintenance operations.
 29. A method according to claim 26wherein determining the respective measures of the relative states ofreadiness of the repairable systems comprises providing respectivemeasures of the relative states of readiness of the repairable systemsthat will be operational on the date of the requested system allocationbased upon an intensity function appropriate for the type of processthat describes the probability of failure of the repairable system.